Abstract
In previous work we have defined statistical equilibrium states for 2D incompressible Euler equations. We study here the relaxation process toward equilibrium. This leads to a natural modeling of the small scales in turbulent flows, which might be relevant for meteorological and oceanographic applications. Numerical simulations illustrating the performance of these new models are presented.
Similar content being viewed by others
References
V. Arnold,Les méthodes mathématiques de la mécanique classique (Mir, 1976).
C. Basdevant and R. Sadourny, Modélisation des échelles virtuelles dans la simulation numérique des écoulements turbulents bidimensionnels,J. Mec. Theor. Appl. (NS1983:243–269.
X. J. Carton, G. R. Flierl, and L. M. Polvani, The generation of tripoles from unstable axisymmetric isolated vortex structure,Europhys. Lett. 9(4):339–344 (1989).
X. Carton and B. Legras, The life-cycle of tripoles in two-dimensional incompressible flows,J. Fluid Mech., to appear.
G. Choquet,Lectures on Analysis, Vol. II (Benjamin, New York, 1969).
A. Chorin,Statistical Mechanics and Vortex Motion (American Mathematical Society, Providence, Rhode Island, 1991).
M. A. Denoix, Thesis, Institut de mécanique de Grenoble (1992).
M. A. Denoix, J. Sommeria, and A. Thess, Two-dimensional turbulence: The prediction of coherent structures by statistical mechanics, inProceedings of the 7th Beer-Sheva Seminar on M.H.D. Flows and Turbulence, Jerusalem, 1993, to appear.
T. Dumont, To appear.
J. B. Flor, Coherent vortex structures in stratified fluids. Thesis, Eindhoven (1994).
D. gottlieb and S. A. Orszag,Numerical Analysis of Spectral Methods: Theory and Applications (SIAM, 1977).
E. T. Jaynes, The minimum entropy production principles, InCollected Papers, R. D. Rosenkrantz, ed. (Kluwer, Dordrecht, 1989).
H. Kesten and G. C. Papanicolaou, A limit theorem for turbulent diffusion,Commun. Math. Phys. 65:97–128 (1979).
R. H. Kraichnan, Diffusion by a random velocity field,Phys. Fluids 13:22–31 (1970).
R. Kubo, Stochastic Liouville equation,J. Math. Phys. 4:174–183 (1963).
J. Michel and R. Robert, Large deviations for Young measures and statistical mechanics of infinite dimensional dynamical systems with conservation law,Commun. Math. Phys. 159:195–215 (1994).
J. Michel and R. Robert, Statistical mechanical theory of the great red spot of Jupiter,J. Stat. Phys. 77:645–666 (1994).
J. Miller, Statistical mechanics of Euler equations in two dimensions,Phys. Rev. Lett. 65:2137–2140 (1990).
J. Miller, P. B. Weichman, and M. C. Cross, Statistical mechanics, Euler equations, and Jupiter's red spot,Phys. Rev. A 45:2328–2359 (1992).
A. S. Monin and A. M. Yaglom,Statistical Fluid Mechanics, Vol. I (MIT Press, Cambridge, 1971).
D. Montgomery, W. H. Matthaeus, W. T. Stribling, D. Martinez, and S. Oughton, Relaxation in two dimensions and the “singh-Poisson” equation,Phys. Fluids A 4(1):3–6 (1992).
Y. G. Morel and X. J. Carton, Multipolar vortices in two-dimensional incompressible flows, Preprint, to appear.
L. Onsager, Statistical hydrodynamics,Nuovo Cimento Suppl. 6:279 (1949).
R. Robert, Relaxation towards a statistical equilibrium state in two-dimensional perfect fluid dynamics, inProceedings XIth International Congress of Mathematical Physics (Paris, 1994).
R. Robert, A maximum entropy principle for two-dimensional Euler equations,J. Stat. Phys. 65:531–553 (1991).
R. Robert and J. Sommeria, Statistical equilibrium states for two-dimensional flows,J. Fluid Mech. 229:291–310 (1991).
R. Robert and J. Sommeria, Relaxation towards a statistical equilibrium state in twodimensional perfect fluid dynamics,Phys. Rev. Lett. A 69:2276–2279 (1992).
R. Sadourny, Turbulent diffusion in large scale flows, inLarge-Scale Transport Processes in Oceans and Atmosphere, J. Willebrand and D.L.T. Anderson, eds. (Reidel, Dordrecht, 1986).
J. Sommeria, C. Staquet and R. Robert, Final equilibrium state of a two-dimensional shear layer,J. Fluid Mech. 233:661–689 (1991).
P. Tabeling et al., Experimental study of decaying turbulence,Phys. Rev. Lett. 67: 3772–3775 (1991).
A. Thess, J. Sommeria, and B. Jüttner, Inertial organization of a two-dimensional turbulent vortex street,Phys. Fluids 6(7):2417–2429 (1994).
G. J. F. Van Heijst, R. C. Kloosterziel, and C. W. M. Williams, Laboratory experiments on the tripolar vortex in a rotating fluid,J. Fluid Mech. 225:301–331 (1991).
V. I. Youdovitch, Non-stationary flow of an incompressible liquid,Zh. Vych. Mat. 3:1032–1066 (1963).
M. I. Freidlin and A. D. Wentzell,Random Perturbations of Dynamical Systems (Springer, Berlin, 1984).
P. Constantin and J. Wu, The inviscid limit for non-smooth vorticity, Preprint.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Robert, R., Rosier, C. The modeling of small scales in two-dimensional turbulent flows: A statistical mechanics approach. J Stat Phys 86, 481–515 (1997). https://doi.org/10.1007/BF02199111
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02199111